Final answer:
By calculating the z-score for the given data and using the standard normal distribution, we find that the probability the mean GPA of a random sample of 20 students is 2.88 or lower is approximately 0.0143, closest to Option A.
Step-by-step explanation:
The question involves finding the probability that the mean GPA of a random sample of 20 students is 2.88 or lower, given that the students' GPAs follow a normal distribution with a mean of 3.02 and a standard deviation of 0.29. To solve this, we will use the standard normal distribution (z-distribution) because the sample size is sufficiently large (n ≥ 30 or, for smaller samples, when the population distribution is known to be normal).
To compute the z-score for the sample mean, we use the formula:
z = (μsample− μpopulation) / (σpopulation/sqrt(n))
where μsample is the sample mean, μpopulation is the population mean, σpopulation is the population standard deviation, and n is the sample size.
Substituting the given values:
z = (2.88 - 3.02) / (0.29 / sqrt(20))
z = -0.14 / (0.29 / 4.4721)
z = -2.19
Now we look up the z-score in the standard normal distribution table or use a calculator to find that the probability (P) of having a z-score of -2.19 or lower is approximately 0.0143, which is closest to Option A.